Problem: Solving Absolute Value Equations
The task is to solve the given absolute value equations and verify the provided solutions. Let's go through each equation step by step.
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1. \( |x| = 4 \)
Solution:
The absolute value equation \( |x| = 4 \) means that \( x \) can be either 4 or -4 because the absolute value of a number is its distance from zero on the number line.
\[
|x| = 4 \implies x = 4 \quad \text{or} \quad x = -4
\]
Answer: \( \{4, -4\} \)
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####
2. \( |x - 4| = 0 \)
Solution:
The absolute value of a number is zero only when the number itself is zero. Therefore:
\[
|x - 4| = 0 \implies x - 4 = 0 \implies x = 4
\]
Answer: \( \{4\} \)
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3. \( \left| \frac{x}{2} \right| = 4 \)
Solution:
The equation \( \left| \frac{x}{2} \right| = 4 \) means that \( \frac{x}{2} \) can be either 4 or -4.
\[
\left| \frac{x}{2} \right| = 4 \implies \frac{x}{2} = 4 \quad \text{or} \quad \frac{x}{2} = -4
\]
Solving each case:
- If \( \frac{x}{2} = 4 \):
\[
x = 4 \cdot 2 = 8
\]
- If \( \frac{x}{2} = -4 \):
\[
x = -4 \cdot 2 = -8
\]
Answer: \( \{8, -8\} \)
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####
4. \( |x + 4| = 8 \)
Solution:
The equation \( |x + 4| = 8 \) means that \( x + 4 \) can be either 8 or -8.
\[
|x + 4| = 8 \implies x + 4 = 8 \quad \text{or} \quad x + 4 = -8
\]
Solving each case:
- If \( x + 4 = 8 \):
\[
x = 8 - 4 = 4
\]
- If \( x + 4 = -8 \):
\[
x = -8 - 4 = -12
\]
Answer: \( \{4, -12\} \)
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####
5. \( |2x - 3| = 9 \)
Solution:
The equation \( |2x - 3| = 9 \) means that \( 2x - 3 \) can be either 9 or -9.
\[
|2x - 3| = 9 \implies 2x - 3 = 9 \quad \text{or} \quad 2x - 3 = -9
\]
Solving each case:
- If \( 2x - 3 = 9 \):
\[
2x = 9 + 3 = 12 \implies x = \frac{12}{2} = 6
\]
- If \( 2x - 3 = -9 \):
\[
2x = -9 + 3 = -6 \implies x = \frac{-6}{2} = -3
\]
Answer: \( \{6, -3\} \)
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6. \( |x - 7| = |x + 1| \)
Solution:
The equation \( |x - 7| = |x + 1| \) means that the distances of \( x \) from 7 and -1 are equal. This occurs when \( x \) is equidistant from 7 and -1. The midpoint between 7 and -1 is:
\[
\text{Midpoint} = \frac{7 + (-1)}{2} = \frac{6}{2} = 3
\]
Thus, \( x = 3 \).
To verify:
- If \( x = 3 \):
\[
|3 - 7| = |3 + 1| \implies |-4| = |4| \implies 4 = 4
\]
Answer: \( \{3\} \)
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####
7. \( |3x - 4| = |2x + 5| \)
Solution:
The equation \( |3x - 4| = |2x + 5| \) implies that either:
1. \( 3x - 4 = 2x + 5 \)
2. \( 3x - 4 = -(2x + 5) \)
Case 1: \( 3x - 4 = 2x + 5 \)
\[
3x - 4 = 2x + 5 \implies 3x - 2x = 5 + 4 \implies x = 9
\]
Case 2: \( 3x - 4 = -(2x + 5) \)
\[
3x - 4 = -2x - 5 \implies 3x + 2x = -5 + 4 \implies 5x = -1 \implies x = -\frac{1}{5}
\]
Answers: \( \{9, -\frac{1}{5}\} \)
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Final Answers:
\[
\boxed{
\begin{array}{ll}
1. & \{4, -4\} \\
2. & \{4\} \\
3. & \{8, -8\} \\
4. & \{4, -12\} \\
5. & \{6, -3\} \\
6. & \{3\} \\
7. & \{9, -\frac{1}{5}\}
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of algebra 2 absolute value equations worksheet.
